In a large pot, combine flour, melted butter, eggs and a pinch of salt and mix well with the milk. Mix until the batter becomes smooth and not too thick. Add more milk if necessary.
Bring saltwater to boil in a large pot.Using a spaezle slicer, let the batter drip directly into the boiling water (otherwise the dumplings will stick together). As soon as the dumplings begin to float at the top of the water, drain them and douse with cold water. Dribble in a few drops of oil, and stir through so the dumplings don’t stick together.
Toss the dumplings well in a pan with butter. Season the eggs with salt, pepper and nutmeg and stir. Pour over the dumplings and cook for no more than two minutes. Serve the egg dumplings on a warmed plate and sprinkle with chives.